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|
; Fully Ordered Finite Sets
; Copyright (C) 2003-2012 Kookamara LLC
;
; Contact:
;
; Kookamara LLC
; 11410 Windermere Meadows
; Austin, TX 78759, USA
; http://www.kookamara.com/
;
; License: (An MIT/X11-style license)
;
; Permission is hereby granted, free of charge, to any person obtaining a
; copy of this software and associated documentation files (the "Software"),
; to deal in the Software without restriction, including without limitation
; the rights to use, copy, modify, merge, publish, distribute, sublicense,
; and/or sell copies of the Software, and to permit persons to whom the
; Software is furnished to do so, subject to the following conditions:
;
; The above copyright notice and this permission notice shall be included in
; all copies or substantial portions of the Software.
;
; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
; IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
; FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
; AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
; LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
; DEALINGS IN THE SOFTWARE.
;
; Original author: Jared Davis <jared@kookamara.com>
; outer.lisp
;
; Theorems relating the more complicated set operations (union, intersect,
; etc.) to one another.
(in-package "SET")
(include-book "delete")
(include-book "union")
(include-book "intersect")
(include-book "difference")
(include-book "cardinality")
(set-verify-guards-eagerness 2)
#||
Q. Why do we need SUBSET enabled?
I think I understand what's going on here. For a reproducible example, you can
just open up std/osets/outer.lisp and try to prove the first theorem with
subset disabled. This leads to the goal below, which is perfectly fine, just
before the pick-a-point strategy kicks in:
Subgoal 2.2
(IMPLIES (IN A Y)
(SUBSET (UNION Y (DELETE A X))
(UNION X Y))).
But after the hint applies, it produces a really weird goal, which ultimately
fails to prove and seems to be the real culprit:
Subgoal 2.2.1
(EQUAL (SUBSET SET-FOR-ALL-REDUCTION (UNION X Y))
(COND ((EMPTY SET-FOR-ALL-REDUCTION) T)
((IN (HEAD SET-FOR-ALL-REDUCTION)
(UNION X Y))
(SUBSET (TAIL SET-FOR-ALL-REDUCTION)
(UNION X Y)))
(T NIL))).
I don't like this goal at all. It appears to be due to needing to prove that
(SUBSET X Y) is equivalent to (ALL X) when (PREDICATE A) == (IN A Y). The goal
above basically corresponds to the recursive definition of ALL:
(defun all (set-for-all-reduction)
(declare (xargs :guard (setp set-for-all-reduction)))
(if (empty set-for-all-reduction)
t
(and (predicate (head set-for-all-reduction))
(all (tail set-for-all-reduction)))))
After tinkering around with several solutions, I think I found one that will
work well. I'm doing a full regression with it now. I'm not planning to
disable subset by default (which would be a bigger change), but with this rule
in place it does seem like the pick-a-point proofs in the osets library are
going through OK even with subset disabled, so I'm optimistic that the new
rule will work:
(defthm pick-a-point-subset-constraint-helper
;; When we do a pick-a-point proof of subset, we need to show that (SUBSET X
;; Y) is just the same as (ALL X) with (PREDICATE A) = (IN A Y). Since ALL
;; is defined recursively, the proof goals we get end up mentioning
;; HEAD/TAIL. This doesn't always work well if the user's theory doesn't
;; have the right rules enabled. This rule is intended to open up SUBSET in
;; only this very special case to solve such goals.
(implies (syntaxp (equal set-for-all-reduction 'set-for-all-reduction))
(equal (subset set-for-all-reduction rhs)
(cond ((empty set-for-all-reduction) t)
((in (head set-for-all-reduction) rhs)
(subset (tail set-for-all-reduction) rhs))
(t nil)))))
Once the regressions pass I'll push it...
Cheers,
Jared
||#
(defthm union-delete-X
(equal (union (delete a X) Y)
(if (in a Y)
(union X Y)
(delete a (union X Y)))))
(defthm union-delete-Y
(equal (union X (delete a Y))
(if (in a X)
(union X Y)
(delete a (union X Y)))))
(defthm intersect-delete-X
(equal (intersect (delete a X) Y)
(delete a (intersect X Y))))
(defthm intersect-delete-Y
(equal (intersect X (delete a Y))
(delete a (intersect X Y))))
(defthm union-over-intersect
(equal (union X (intersect Y Z))
(intersect (union X Y) (union X Z))))
(defthm intersect-over-union
(equal (intersect X (union Y Z))
(union (intersect X Y) (intersect X Z))))
(defthm difference-over-union
(equal (difference X (union Y Z))
(intersect (difference X Y) (difference X Z))))
(defthm difference-over-intersect
(equal (difference X (intersect Y Z))
(union (difference X Y) (difference X Z))))
(defthm difference-delete-X
(equal (difference (delete a X) Y)
(delete a (difference X Y))))
(defthm difference-delete-Y
(equal (difference X (delete a Y))
(if (in a X)
(insert a (difference X Y))
(difference X Y))))
(defthm difference-insert-Y
(equal (difference X (insert a Y))
(delete a (difference X Y))))
(defthm intersect-cardinality-X
(<= (cardinality (intersect X Y))
(cardinality X))
:rule-classes (:rewrite :linear))
(defthm intersect-cardinality-Y
(<= (cardinality (intersect X Y))
(cardinality Y))
:rule-classes (:rewrite :linear))
; There are also some interesting properties about cardinality which are more
; precise.
(defthm expand-cardinality-of-union
;; This is pretty questionable -- it used to also be a :linear rule, but that was
;; really expensive.
(equal (cardinality (union X Y))
(- (+ (cardinality X) (cardinality Y))
(cardinality (intersect X Y)))))
(defthm expand-cardinality-of-difference
;; Also questionable, also used to be :linear
(equal (cardinality (difference X Y))
(- (cardinality X)
(cardinality (intersect X Y)))))
;; We used to have this rule, but it was silly -- (intersect X Y) can just rewrite to
;; (SFIX X) when X is a subset of Y.
;; (defthm intersect-cardinality-subset
;; (implies (subset X Y)
;; (equal (cardinality (intersect X Y))
;; (cardinality X))))
(defthmd intersect-cardinality-non-subset
(implies (not (subset x y))
(< (cardinality (intersect x y))
(cardinality x)))
:rule-classes (:rewrite :linear)
:hints(("Goal" :in-theory (enable subset))))
(defthmd intersect-cardinality-subset-2
(equal (equal (cardinality (intersect X Y))
(cardinality X))
(subset X Y)))
(defthmd intersect-cardinality-non-subset-2
(equal (< (cardinality (intersect x y))
(cardinality x))
(not (subset x y))))
|
